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IE111_F09_HW03_soln - IE 111 Fall Semester 2009 Homework#3...

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IE 111 Fall Semester 2009 Homework #3 Solutions 1) Samples of skin experiencing desquamation are analyzed for both moisture and melanin content. The results from 100 skin samples are as follows: Melanin content High low Moisture high 13 7 Content low 48 32 Let A denote the event that a sample has low melanin content, and let B denote the event that sample has high moisture content. Determine the following probabilities: a) P(A)=(7+32)/100=0.39 b) P(B)=(13+7)/100=0.2 c) P(A|B)=7/20=0.35 d) P(B|A)=7/39=0.179 2) An integer is selected at random from {1, 2, …, 100}. Given that the number selected is divisible by 2, find the probability that it is divisible by 3 or 5. Solution: Let A_2 = event that the number is divisible by 2 Let A_3 = event that the number is divisible by 3 Let A_5 = event that the number is divisible by 5 P(A_3 A_5 |A_2) = P[(A_3 A_5) A_2 ] / P(A_2) = P [(A_3 A_2) (A_5 A_2) ] / P(A_2) = P(A_3 A_2) + P (A_5 A_2)–P(A_3 A_5 A_2)/ P(A_2) P(A_3 A_2) = 16/100, P (A_5 A_2) = 10/100, P(A_3 A_5 A_2)= 3/100 P(A_3 A_5 | A_2)= (16/100 + 10/100 + 3/100) / (50/100) = 23/50 = 0.46 1
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3) We have two coins, the first is fair and the second two-headed. We pick one of the coins at random, we toss it twice and heads shows both times. Find the probability that the coin picked is fair. Solution: In this experiment, each outcome consists of a particular coin selection and two coin tosses. So for example, fhh represents picking the fair coin and obtaining hh. Now, the event F ={the selected coin is fair}={fhh,fht, fth,ftt}, F c = {the selected coin is two-headed}={f c hh} P(F)=P(F c ) =1/2 P(hh|F)=1/4, p(hh|F c )=1 We want to find P (F|hh), so P(hh)= P(hh|F)P(F) + P(hh|F c )P(F c ) =5/8 P(F/hh)= P(hh/F)P(F)/ P(hh) = (1/4) (1/2) / (5/8) =1/5 4) A lot of 50 semiconductors chips contains 10 that are defective. Two chips are selected at random, without replacement, from the lot. a) What is the probability that the first one selected is defective?
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